Algebrability and nowhere Gevrey differentiability Bastin, Françoise ; ; Esser, Céline et al in Israel Journal of Mathematics (in press) We show that there exist c-generated algebras (and dense in C^infty([0,1])) every nonzero element of which is a nowhere Gevrey diff erentiable function. This leads to results of dense algebrability (and ... [more ▼] We show that there exist c-generated algebras (and dense in C^infty([0,1])) every nonzero element of which is a nowhere Gevrey diff erentiable function. This leads to results of dense algebrability (and, therefore, lineability) of functions enjoying this property. In the process of proving these results we also provide a new construction of nowhere Gevrey di fferentiable functions. [less ▲] Detailed reference viewed: 77 (32 ULg)Regularity of functions: Genericity and multifractal analysis Esser, Céline Doctoral thesis (2014) As surprising as it may seem, there exist functions of C∞(R) which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A ... [more ▼] As surprising as it may seem, there exist functions of C∞(R) which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A classical technique is to use the Baire category theorem and the notion of residuality. This notion is purely topological and does not give any information about the measure of the set of objects satisfying such a property. In this purpose, the notion of prevalence has been introduced. Moreover, one could also wonder whether large algebraic structures of such objects can be constructed. This question is formalized by the notion of lineability. The first objective of the thesis is to go further into the study of nowhere analytic functions. It is known that the set of nowhere analytic functions is residual and lineable in C∞([0, 1]). We prove that the set of nowhere analytic functions is also prevalent in C∞([0, 1]). Those results of genericity are then generalized using Gevrey classes, which can be seen as intermediate between the space of analytic functions and the space of infinitely differentiable functions. We also study how far such results of genericity could be extended to spaces of ultradifferentiable functions, defined using weight sequences or using weight functions. The second main objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in most of the practical cases, but there exist heuristic methods, called multifractal formalisms, which allow to estimate this spectrum and which give satisfactory results in many situations. The Frisch-Parisi conjecture, classically used and based on Besov spaces, presents two disadvantages: it can only hold for spectra that are concave and it can only yield the increasing part of spectra. Concerning the first problem, the use of Snu spaces allows to deal with non-concave increasing spectra. Concerning the second problem, a generalization of the Frisch-Parisi conjecture obtained by replacing the role played by wavelet coefficients by wavelet leaders allows to recover the decreasing part of concave spectra. Our purpose in this thesis is to combine both approaches and define a new formalism derived from large deviations based on statistics of wavelet leaders. As expected, we show that this method yields non-concave spectra and is not limited to their increasing part. From the theoretical point of view, we prove that this formalism is more efficient than the previous wavelet-based multifractal formalisms. We present the underlying function space and endow it with a topology. [less ▲] Detailed reference viewed: 22 (9 ULg)Detection of non-concave and non-increasing spectra: Snu spaces revisited with wavelet leaders Esser, Céline Conference (2014, September 26) Our objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in ... [more ▼] Our objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in most of the practical cases, but there exist heuristic methods, called multifractal formalisms, which allow to estimate this spectrum and which give satisfactory results in many situations. The Frisch-Parisi conjecture, classically used and based on Besov spaces, presents two disadvantages: it can only hold for spectra that are concave and it can only yield the increasing part of spectra. Concerning the first problem, the use of S spaces allows to deal with non-concave increasing spectra. Concerning the second problem, a generalization of the Frisch-Parisi conjecture obtained by replacing the role played by wavelet coefficients by wavelet leaders allows to recover the decreasing part of concave spectra. We present a combination of both approaches to define a new formalism derived from large deviations based on statistics of wavelet leaders. We also present the associated function space. [less ▲] Detailed reference viewed: 10 (0 ULg)Genericity and classes of ultradifferentiable functions Esser, Céline Conference (2014, September 26) As surprising as it may seem, there exist infinitely differentiable functions which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones ... [more ▼] As surprising as it may seem, there exist infinitely differentiable functions which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A classical technique is to use the Baire category theorem and the notion of residuality. This notion is purely topological and does not give any information about the measure of the set of objects satisfying such a property. In this purpose, the notion of prevalence has been introduced. Moreover, one could also wonder whether large algebraic structures of such objects can be constructed. This question is formalized by the notion of lineability. The first objective of this talk is to go further into the study of nowhere analytic functions. It is known that the set of nowhere analytic functions is residual and lineable in C^infty([0, 1]). We prove that the set of nowhere analytic functions is also prevalent in this space. Those results of genericity are then generalized using Gevrey classes, which can be seen as intermediate between the space of analytic functions and the space of infinitely differentiable functions. We also study how far such results of genericity could be extended to spaces of ultradifferentiable functions, defined using weight sequences. [less ▲] Detailed reference viewed: 12 (0 ULg)Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I) Esser, Céline ; Kleyntssens, Thomas ; Bastin, Françoise et al Conference (2014, May 22) Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to ... [more ▼] Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdor ff dimension of the set of points which have the same H ölder exponent. For real-life signals, the computation of the spectrum of singularities from its de finition is not feasible. Multifractal formalisms are used to approximate this spectrum. Currently, there exist several methods. In this talk, we present a new multifractal formalism based on the wavelet leaders of a signal which allows to detect non concave and non increasing spectra. [less ▲] Detailed reference viewed: 25 (6 ULg)Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part II) Kleyntssens, Thomas ; Esser, Céline ; Nicolay, Samuel Conference (2014, May 22) This talk follows "Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I)" given by Céline Esser. A multifractal formalism is a numerically computable formula that ... [more ▼] This talk follows "Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I)" given by Céline Esser. A multifractal formalism is a numerically computable formula that approximates the spectrum of singularities of a function. A new multifractal formalism based on the wavelet leaders is presented as well as a comparison with other formalisms. Its main advantages are that it allows to detect non concave and non increasing spectra. An implementation is proposed. [less ▲] Detailed reference viewed: 25 (12 ULg)A new multifractal formalism based on wavelet leaders : detection of non concave and non increasing spectra (Part I) Esser, Céline ; Kleyntssens, Thomas ; Nicolay, Samuel et al Conference (2014, March 25) Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to ... [more ▼] Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdorff dimension of the set of points which have the same Hölder exponent. The spectrum of singularities of many mathematical functions can be determined directly from its definition. However, for many real-life signals, the numerical determination of their Hölder regularity is not feasible. Therefore, one cannot expect to have a direct access to their spectrum of singularities and one has to find an indirect way to compute it. A multifractal formalism is a formula which is expected to yield the spectrum of singularities from quantities which are numerically computable. Several multifractal formalisms based on the wavelet coefficients of a signal have been proposed to estimate its spectrum. The most widespread of these formulas is the so-called thermodynamic multifractal formalism, based on the Frish-Parisi conjecture. This formalism presents two drawbacks: it can hold only for spectra that are concave and it can yield only the increasing part of the spectrum. This first problem can be avoided using Snu spaces. The second one can be avoided using a formalism based on wavelet leaders of the signal. In this talk, we propose a new multifractal formalism, based on a generalization of the Snu spaces using wavelet leaders. It allows to detect non concave and non increasing spectra. An implementation of this method is presented in the talk "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)" of T. Kleyntssens. [less ▲] Detailed reference viewed: 38 (11 ULg)A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II) Kleyntssens, Thomas ; Esser, Céline ; Nicolay, Samuel Conference (2014, March 25) This talk follows "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part I)" given by Céline Esser. For real-life signals, it is impossible to ... [more ▼] This talk follows "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part I)" given by Céline Esser. For real-life signals, it is impossible to compute the spectrum of singularities by using its definition. A multifractal formalism is used to approximate this spectrum. We present a new multifractal formalism for non concave and non increasing spectra based on wavelet leaders. In this talk, an implementation of this formalism is given and several numerical examples are presented. [less ▲] Detailed reference viewed: 35 (14 ULg)Revisiting Snu spaces with wavelet leaders to detect non concave and non increasing spectra Esser, Céline Poster (2014, February 27) A multifractal formalism is a formula which is expected to yield the spectrum of singularities of a function from quantities which are numerically computable. The most widespread of these formulas is the ... [more ▼] A multifractal formalism is a formula which is expected to yield the spectrum of singularities of a function from quantities which are numerically computable. The most widespread of these formulas is the so-called thermodynamic mul- tifractal formalism, based on the Frish-Parisi conjecture. It presents two drawbacks: it can hold only for spectra which are concave and it can only yield the increasing part of the spectrum. This first problem can be avoided using Sν spaces. The second one can be taken care using the wavelet leaders method. In this poster, we present a new multifractal formalism based on a generaliza- tion of the Sν spaces using wavelet leaders. It allows to detect non concave and non increasing spectra. We compare this formalism with the Sν method and the wavelet leaders method. It is based on joint works with F. Bastin, S. Jaffard, T. Kleyntssens and S. Nicolay. [less ▲] Detailed reference viewed: 42 (21 ULg)Generic results in classes of ultradifferentiable functions Esser, Céline in Journal of Mathematical Analysis & Applications (2014), 413(1), 378391 Let E be a Denjoy Carleman class of ultradifferentiable functions of Beurling type on the real line that strictly contains another class F of Roumieu type. We show that the set S of functions in E that ... [more ▼] Let E be a Denjoy Carleman class of ultradifferentiable functions of Beurling type on the real line that strictly contains another class F of Roumieu type. We show that the set S of functions in E that are nowhere in the class F is large in the topological sense (it is residual), in the measure theoretic sense (it is prevalent), and that SU{0} contains an infinite dimensional linear subspace (it is lineable). Consequences for the Gevrey classes are given. Similar results are also obtained for classes of ultradifferentiable functions defined imposing conditions on the Fourier Laplace transform of the function. [less ▲] Detailed reference viewed: 42 (19 ULg)Algebrability of nowhere Gevrey differentiable functions Esser, Céline Conference (2013, September 17) Detailed reference viewed: 30 (3 ULg)Generic results in Denjoy-Carleman classes Esser, Céline Poster (2013, September 09) Denjoy-Carleman classes are spaces of smooth functions which satisfy growth conditions on their derivatives. We distinguish the class of ultradi fferentiable functions of Roumieu type and the class of ... [more ▼] Denjoy-Carleman classes are spaces of smooth functions which satisfy growth conditions on their derivatives. We distinguish the class of ultradi fferentiable functions of Roumieu type and the class of ultradi fferentiable functions of Beurling typ. Endowed with its natural topology, the Beurling class is a Fr échet space. In the poster, we give a condition to have the strict inclusion of a Roumieu class into aBeurling class. We obtain then generic results about the set of functions of a Beurling class which are nowhere in a Roumieu class. Those generic results are obtained from three di fferent points of view: using the Baire category theorem, the notion of prevalence and the notion of lineability. We also study the particular case of Gevrey classes. [less ▲] Detailed reference viewed: 27 (9 ULg)Prevalence of nowhere analytic functions Esser, Céline Conference (2013, May 30) Detailed reference viewed: 34 (11 ULg)Applications of the concept of Prevalence: Nowhere analytic functions and Multifractal analysis Esser, Céline Conference (2013, February 07) Detailed reference viewed: 33 (10 ULg)Applications of the concept of prevalence: Nowhere analytic functions and multifractal analysis Esser, Céline Conference (2013, February 05) Detailed reference viewed: 19 (11 ULg)About Generic Properties of "Nowhere Analyticity" Bastin, Françoise ; Nicolay, Samuel ; Esser, Céline Conference (2012, May 08) A infinitely differentiable function f is is analytic at a point x if its Taylor series at this point converges to f on an open neighbourhood of x; if this is not the case, f has a singularity at x. A ... [more ▼] A infinitely differentiable function f is is analytic at a point x if its Taylor series at this point converges to f on an open neighbourhood of x; if this is not the case, f has a singularity at x. A function with a singularity at each point of the interval is called nowhere analytic on the interval. In this talk, we show that the set of nowhere analytic functions is prevalent in the Frechet space C([0;1]). We get then a deeper result using Gevrey classes. [less ▲] Detailed reference viewed: 39 (20 ULg)Prevalence of ''nowhere analyticity'' Bastin, Françoise ; Esser, Céline ; Nicolay, Samuel in Studia Mathematica (2012), 210(3), This note brings a complement to the study of genericity of functions which are nowhere analytic mainly in a measure-theoretic sense. We extend this study in Gevrey classes of functions. Detailed reference viewed: 50 (23 ULg)Prevalence of ''nowhere analyticity'' Bastin, Françoise ; Esser, Céline ; Nicolay, Samuel in Studia Mathematica (2012), 210(3), This note brings a complement to the study of genericity of functions which are nowhere analytic mainly in a measure-theoretic sense. We extend this study in Gevrey classes of functions. Detailed reference viewed: 50 (23 ULg)Prevalence of ''nowhere analyticity'' Bastin, Françoise ; Esser, Céline ; Nicolay, Samuel in Studia Mathematica (2012), 210(3), This note brings a complement to the study of genericity of functions which are nowhere analytic mainly in a measure-theoretic sense. We extend this study in Gevrey classes of functions. Detailed reference viewed: 50 (23 ULg)Bastin, Françoise ; Esser, Céline ; Nicolay, Samuel in Studia Mathematica (2012), 210(3), |
||